An \(ABC\) inequality for Mahler's measure
Let \(\overline{\mathbb Q}\) be an algebraic closure of \(\mathbb Q\), and for \(N \in \mathbb N\), let \(\phi_0, \phi_1, \dots, \phi_N \in \overline{\mathbb Q} [x]\) be polynomials with no common zeroes, satisfying \[ \phi_0 + \phi_1 + \dots + \phi_N =0, \] and spanning a \(\overline{\mathbb Q}\)-vector space of dimension \(N\) inside \(\overline{\mathbb Q} [x]\). Let \(\mathcal Z\) denote the set of all zeroes of \(\prod_{i=0}^N \phi_i\) in \(\overline{\mathbb Q}\). It is proved that \[ \log \mu(\phi_0) \leq \binom N 2 \sum_{\alpha \in \mathcal Z} \log H(\alpha) + \Delta_N (\deg \phi_1, \dots, \deg \phi_N), \] where \(\mu\) denotes the global Mahler measure of a polynomial, \(H\) the absolute Weil height of an algebraic number, and \(\Delta_N\) the maximal local logarithmic Mahler measure of the Wronskian of \(N\) normed polynomials of fixed degrees, where the roots vary on the unit circle. Only \(\Delta_2 (m,n) = \log(m+n)\) is explicitly known, for \(N \geq 3\) upper bounds for \(\Delta_N\) are given in Lemmas 4 and 5 of the paper. This result can be seen as an analogue of Mason's \(abc\)-theorem for Mahler's measure. The proof runs similar to that of the generalized \(abc\)-theorem for \(N+1\) summands, see e.g. \textit{W. D. Brownawell} and \textit{D. W. Masser} [Math. Proc. Camb. Philos. Soc. 100, 427--434 (1986; Zbl 0612.10010)]. A special challenge is the new analytical problem of estimating \(\Delta_N\).
scientific article; zbMATH DE number 5313121
scientific article